On the stochastic and dependence properties of the three-state systems
Suppose that a system has three states up, partial performance and down. We assume that for a random time <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$T_1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>T</mi> <mn>1</mn> </msub> </math> </EquationSource> </InlineEquation> the system is in state up, then it moves to state partial performance for time <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$T_2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>T</mi> <mn>2</mn> </msub> </math> </EquationSource> </InlineEquation> and then the system fails and goes to state down. We also denote the lifetime of the system by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$T$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>T</mi> </math> </EquationSource> </InlineEquation>, which is clearly <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$T=T_1+T_2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>T</mi> <mo>=</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>T</mi> <mn>2</mn> </msub> </mrow> </math> </EquationSource> </InlineEquation>. In this paper, several stochastic comparisons are made between <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$T$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>T</mi> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$T_1$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>T</mi> <mn>1</mn> </msub> </math> </EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$T_2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>T</mi> <mn>2</mn> </msub> </math> </EquationSource> </InlineEquation> and their reliability properties are also investigated. We prove, among other results, that different concepts of dependence between the elements of the signatures (which are structural properties of the system) are preserved by the lifetimes of the states of the system (which are aging properties of the system). Various illustrative examples are provided. Copyright Springer-Verlag Berlin Heidelberg 2015
